{"id":"openai-math/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space","pdf":"https://github.com/openai/math/blob/main/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026","tags":["differential-geometry","isometric-embedding","riemannian-geometry"],"score":0,"title":"A Smooth Metric with No Local Isometric Immersion into Three-Space","domain":"Differential geometry","family":"334","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"We construct a smooth positive-definite Riemannian metric on $`(-1,1)^2`$ that agrees with the Euclidean metric to every order at the origin, yet no neighborhood of the origin admits a smooth isometric immersion into Euclidean three-space. This gives a negative answer to the unrestricted smooth local isometric realization problem for surfaces.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":true,"family_title":"A smooth surface metric with no local isometric immersion in ℝ 3"}